# Two-point Form of a Line

We will discuss here about the method of finding the equation of a straight line in the two point form.

To find the equation of a straight line in the two point form,

Let AB be a line passing through two points A (x$$_{1}$$, y$$_{1}$$) and B (x$$_{2}$$, y$$_{2}$$).

Let the equation of the line be y = mx + c ................... (i), where m is the slope of the line and c is the y-intercept.

As (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) are points on the line AB, (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) satisfy (i).

Therefore, y$$_{1}$$ = mx$$_{1}$$ + c ................................ (ii)

and y$$_{2}$$ = mx$$_{2}$$ + c ................................ (iii)

Subtracting (iii) from (ii),

y$$_{1}$$ - y$$_{2}$$ = m(x$$_{1}$$ - x$$_{2}$$)

⟹ m = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$ ................................ (iv)

Substituting m = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$ in (ii),

y$$_{1}$$ = [$$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$]x$$_{1}$$ + c

⟹ c = y$$_{1}$$ - $$\frac{x_{1}(y_{1} - y_{2})}{ x_{1} - x_{2}}$$

c = $$\frac{ y_{1}(x_{1} - x_{2}) - x_{1}(y_{1} - y_{2})}{ x_{1} - x_{2}}$$

c = $$\frac{x_{1}y_{2} - x_{2}y_{1}}{ x_{1} - x_{2}}$$

Therefore, from (i),

y = [$$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$]x + $$\frac{x_{1}y_{2} - x_{2}y_{1}}{ x_{1} - x_{2}}$$

Subtracting y$$_{1}$$ from both sides of (v)

y - y$$_{1}$$ = [$$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$]x + $$\frac{x_{1}y_{2} - x_{2}y_{1}}{ x_{1} - x_{2}}$$

y - y$$_{1}$$ = [$$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$]x + $$\frac{x_{1}(y_{2} - y_{1})}{ x_{1} - x_{2}}$$

y - y$$_{1}$$ = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$(x + x$$_{1}$$)

The equation of the straight line passing through (x1, y1) and (x2, y2) is y - y$$_{1}$$ = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$(x + x$$_{1}$$)

Note: From (iv), the slope of the line joining the points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) is $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$ i.e., $$\frac{Difference of y-coordinates}{difference of x-coordinates in the same order}$$

Solved example on two-point form of a line:

The equation of the line passing through the points (1, 1) and (-3, 2) is

y - 1 = $$\frac{1 - 2}{1 - (-3)}$$(x - 1)

⟹ y – 1 = -$$\frac{1}{4}$$(x – 1)

Also, y – 2 = $$\frac{2 - 1}{-3 - 1}$$(x + 3)

⟹ y – 2 = -$$\frac{1}{4}$$(x + 3)

However, the two equations are the same.

Equation of a Straight Line